#### Translator’s note

Douady, A. Séminaire Henri Cartan 13 (1) (1960–61), Talks no. 2 (“Variétés et espaces mixtes”), 3 (“Déformations régulières”), and 4 (“Obstruction primaire à la déformation”). http://www.numdam.org/item/SHC_1960-1961__13_1

The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.

Version: 367a5c6

[Translator] According to the complete list of talks, the notes from the first talk of the 1960/61 Séminaire Henri Cartan — “Fibrés en tores complexes” (also given by Adrien Douady) — were not copied, and thus seem to be lost to the past. What follows is a translation of the next three talks in this seminar series.

# I. Category of models

Let B be a topological space. We define the category \mathscr{S}_B^n in the following manner: the objects of \mathscr{S}_B^n are the open subsets of B\times\mathbb{C}^n, and a morphism f\colon U\to U' from an open subset U\subset B\times\mathbb{C}^n to an open subset U'\subset B\times\mathbb{C}^n is a continuous map f\colon U\to U' satisfying the following two conditions:

1. the diagram \begin{CD} U @>f>> U' \\@V{\pi_1}VV @VV{\pi_1}V \\B @= B \end{CD} commutes, where \pi_1 denotes the projection of B\times\mathbb{C}^n to B; and
2. for all x\in B, the map f_x\colon U_x\to U'_x is holomorphic, where U_x = \{z\in\mathbb{C}^n \mid (x,z)\in U\} (and similarly for U').

If B is endowed with the structure of a \mathscr{C}^\infty manifold (resp. an \mathbb{R}-analytic manifold, resp. \mathbb{C}-analytic manifold), then we obtain a category \mathscr{C}^\infty\mathscr{S}_B (resp. \mathbb{R}\mathscr{S}_B, resp. \mathbb{C}\mathscr{S}_B) by requiring the morphisms to be \mathscr{C}^\infty (resp. \mathbb{R}-analytic, resp. \mathbb{C}-analytic).

More generally, if f_1\colon B\to B' is a continuous map from one topological space to another, then a morphism of \mathscr{S}_{f_1} is a continuous map f from an object U of \mathscr{S}_B to an object U' of \mathscr{S}_{B'} such that

1. the diagram \begin{CD} U @>f>> U' \\@V{\pi_1}VV @VV{\pi_1}V \\B @>>{f_1}> B' \end{CD} commutes; and
2. f_x\colon U_x\to U'_{f_1(x)} is holomorphic for all x\in B.

If f_1 is a \mathscr{C}^\infty map from one \mathscr{C}^\infty manifold to another, then f will be a morphism of \mathscr{C}^\infty\mathscr{S}_{f_1} if, further, it is a \mathscr{C}^\infty map (resp. …). We thus obtain, for every category of topological spaces, a fibred category \mathscr{S}^n (resp. \mathscr{C}^\infty\mathscr{S}^n, resp. …).

# II. The definition of mixed spaces and mixed varieties

## 1. First definition

Let B and V be separated spaces, and let \pi\colon V\to B be a continuous map. The structure of a mixed space over B is defined on V by a system of charts \varphi_i\colon U_i\to V, where the (U_i) are objects of \mathscr{S}_B^n; for each i, \varphi_i is a homeomorphism from U_i to an open subset of V such that the diagram \begin{CD} U_i @>{\varphi_i}>> V \\@V{\pi_1}VV @VV{\pi}V \\B @= B \end{CD} commutes; finally, for all i and all j, the “change of chart” \varphi_j^{-1}\circ\varphi_i is an isomorphism of \mathscr{S}_B from an open subset of U_i to an open subset of U_j.

The structure thus defined is that of a (\mathscr{C}^0,\mathbb{C})-mixed space. If B is a \mathbb{C}-analytic space, and if the change of chart maps are all \mathbb{C}-analytic, then we have a \mathbb{C}-analytic mixed space. In this case, V itself is a \mathbb{C}-analytic space, and the fibres V_x=\pi^{-1}(x) are \mathbb{C}-analytic sub-manifolds.

If B is a \mathscr{C}^\infty manifold (resp. \mathbb{R}-analytic, resp. \mathbb{C}-analytic), and if the change of chart maps are all \mathscr{C}^\infty (resp. …), then we have a (\mathscr{C}^\infty,\mathbb{C})-mixed manifold (resp. (\mathbb{R},\mathbb{C}), resp. (\mathbb{C},\mathbb{C})). In this case, V itself is a manifold. Note that the notion of a (\mathbb{C},\mathbb{C})-mixed manifold, or a \mathbb{C}-analytic mixed manifold, reduces to simply having a \mathbb{C}-analytic manifold V endowed with a projection \pi\colon V\to B onto another \mathbb{C}-analytic manifold such that \pi is of maximal rank at every point.1

Let \pi\colon V\to B and \pi'\colon V'\to B' be mixed spaces, and let f_1\colon B\to B' be a continuous (resp. …) map. Then a morphism from V to V' over f_1 is a continuous map f\colon V\to V' such that the diagram \begin{CD} V @>f>> V' \\@V{\pi}VV @VV{\pi'}V \\B @>>{f_1}> B' \end{CD} commutes, and such that, for any charts \varphi_i\colon U_i\to V and \varphi'_j\colon U'_j\to V', the map {\varphi'_j}^{-1}\circ f\circ\varphi_i is a morphism of \mathscr{S}_{f_1} (resp. …) from an open subset of U_i to U_j.

## 2. An equivalent definition

We now give another way of defining mixed spaces, equivalent to the above.

Given separated spaces B and V, along with a continuous map \pi\colon V\to B, the structure of a pre-mixed space consists of the structure of a \mathbb{C}-analytic manifold on each fibre V_x=\pi^{-1}(x). Given pre-mixed spaces \pi\colon V\to B and \pi'\colon V'\to B', along with a continuous map f_1\colon B\to B', a morphism of pre-mixed spaces over f_1 is a continuous map f\colon V\to V' such that the diagram \begin{CD} V @>f>> V' \\@V{\pi}VV @VV{\pi'}V \\B @>>{f_1}> B' \end{CD} commutes and induces a \mathbb{C}-analytic map on each fibre.

A mixed space is a pre-mixed space \pi\colon V\to B such that every point y\in V admits a neighbourhood W in V that is isomorphic as a pre-mixed space to an open subset of B\times\mathbb{C}^n, via an isomorphism over the identity. The morphisms of mixed spaces are the same: mixed spaces form a full subcategory.

## 3. Deformations

A mixed space \pi\colon V\to B is said to be proper if B is locally compact and the map \pi is proper (i.e. the inverse image of any compact subset is compact). If it is a mixed manifold, then we can show that it is a fibred manifold that is locally trivial with respect to the underlying \mathscr{C}^\infty structure, but the previous talk shows that, in general, any two fibres are not isomorphic as \mathbb{C}-analytic manifolds.

Let V_0 be a compact \mathbb{C}-analytic manifold, B a locally compact space, and b_0\in B. Then a \mathbb{C}-analytic deformation of V_0 over (B,b_0) consists of a proper \mathbb{C}-analytic mixed space \pi\colon V\to B along with an isomorphism of \mathbb{C}-analytic manifolds i\colon V_0\to\pi^{-1}(b_0).

The goal of this seminar is the study, at least local, and an attempt at a classification of, \mathbb{C}-analytic deformations of a given compact \mathbb{C}-analytic manifold V_0.

Let V_0 be a compact \mathbb{C}-analytic manifold. A \mathbb{C}-analytic deformation (\pi\colon V\to B,i\colon V_0\to V) of V_0 is said to be locally complete if, for any other deformation (\pi'\colon V'\to B',i'\colon V_0\to V') of V_0, there exists a neighbourhood B'_1 of b'_0 in B', an analytic map f_1\colon B'_1\to B with f_1(b'_0)\to b_0, and a morphism of \mathbb{C}-analytic mixed spaces f\colon {\pi'}^{-1}(B'_1)\to V over f_1 such that f\circ i'=i. The deformation is said to be locally universal is furthermore the germ of f_1 at b'_0 is determined uniquely by this condition.

It seems that every compact \mathbb{C}-analytic manifold V_0 admits a locally complete \mathbb{C}-analytic deformation, and a locally universal one if the group of automorphisms of V_0 is discrete.

# III. Vector fields

## 1. Study on models

Let B be a space, U an object of \mathscr{S}_B (i.e. an open subset of B\times\mathbb{C}^n), b_0 a point of B, and set U_0=\pi^{-1}(b_0).

A holomorphic field of tangent vectors on U_0 (i.e. a holomorphic map from U_0 to \mathbb{C}^n) is said to be a vertical holomorphic field on U_0. A vertical holomorphic field on U is a continuous (resp. …) map \theta\colon U\to\mathbb{C}^n that induces a vertical holomorphic field on each fibre U_x. If f\colon U\to U' is an isomorphism in \mathscr{S}_B, then the transport f_*\theta of \theta by f is defined by f_*\theta(f(x,z)) = \mathrm{D}_2 f_{x,z}\cdot\theta(x,z) where \mathrm{D}_2 f_{x,z} is the linear map from \mathbb{C}^n to itself that is tangent to f_x at the point z\in U_x. This is again a vertical holomorphic field, since it follows from a Cauchy integral that the matrix \mathrm{D}f_{x,z} depends continuously on the pair (x,z).

Now suppose that B is a \mathscr{C}^\infty manifold, just for simplicity, and let T_0 be the tangent space to B at b_0. A field of tangent vectors to U defined on U_0, i.e. a map \omega\colon U_0\to T_0\times\mathbb{C}^n, is said to be a projectable holomorphic field if \omega(b_0,z)=(t_0,\theta(z)) (where t_0\in T_0 is a vector that does not depend on z, called the projection of the field \omega) and \theta(z) is a holomorphic vector field. If B is a \mathbb{C}-analytic space, possibly with a singularity at b_0, then we give the same definition, but with T_0 then being the Zariski tangent space to B at b_0, i.e. the dual of \mathfrak{m}/\mathfrak{m}^2, where \mathfrak{m} is the ideal of germs at b_0 of holomorphic functions on B that vanish at b_0.

If f\colon U\to U' is an isomorphism of \mathscr{C}^\infty\mathscr{S}_B (resp. …), then then transport f_*\omega is defined by f_*\omega(f(b_0,z)) = \mathrm{D}f_{b_0,z}\omega(b_0,z) where \mathrm{D}f_{b_0,z}\colon T_0\times\mathbb{C}^n\to T_0\times\mathbb{C}^n is now the linear map that is tangent to f at the point (b_0,z). This is a projectable holomorphic field. Indeed, the matrix \mathrm{D}f_{b_0,z} can be written as \begin{pmatrix} I & 0 \\\mathrm{D}_1f & \mathrm{D}_2f \end{pmatrix} and \begin{aligned} \mathrm{D}_1f\colon T &\to \mathbb{C}^n \\\mathrm{D}_2f\colon \mathbb{C}^n &\to \mathbb{C}^n \end{aligned} both depend holomorphically on z (for \mathrm{D}_1f, this follows from the fact that f_x is holomorphic for every x). By setting f_*\omega(b_0,z')=(t_0,\theta'(z')), we have \begin{gathered} \theta'(z') = \mathrm{D}_1f_{b_0,z}(t_0) + \mathrm{D}_2f_{b_0,z}(\omega(z)) \\\text{if }z'=f_{b_0}(z) \end{gathered} which shows that f_*\omega is indeed a projectable holomorphic field.

A projectable holomorphic field on U is a \mathscr{C}^\infty field of vectors tangent to U that induces a projectable holomorphic field on each fibre.

## 2. Vector fields on a mixed manifold

Let \pi\colon V\to B be a (\mathscr{C}^\infty,\mathbb{C})-mixed manifold (resp. …, resp. a \mathbb{C}-analytic mixed space). By transporting along the charts, we define the notions of

• vertical holomorphic fields on an open subset of a fibre;
• vertical holomorphic fields on a open subset of V;
• projectable holomorphic fields on an open subset of a fibre; and
• projectable holomorphic fields on an open subset of V.

Let \xi be a \mathscr{C}^\infty vector field (resp. …) on V. By integrating \xi, we obtain a \mathscr{C}^\infty map, denoted by e^\xi, from an open subset W\subset\mathbb{R}\times V containing \{0\}\times V (resp. \mathbb{C}-analytic map from an open subset W\subset\mathbb{C}\times V) to V, characterised by

1. e^\xi(t_1+t_2,y) = e^\xi(t_1,e^\xi(t_2,y)), with the left-hand side being defined whenever the right-hand side is; and
2. \frac{\partial}{\partial t}e^\xi(t,y)|_{0,y} = \xi(y).

Note that W is a mixed manifold over \mathbb{R}\times B (resp. a mixed space over \mathbb{C}\times B).

For e^\xi\colon W\to V to be a morphism of mixed spaces over the projection \mathbb{R}\times B\to B, it is necessary and sufficient for \xi to be a vertical holomorphic field. For e^\xi\colon W\to V to be a morphism of mixed spaces over a map from an open subset of \mathbb{R}\times B containing \{0\}\times B to B, it is necessary and sufficient for \xi to be a projectable holomorphic field.

The proof is left to the reader.

# IV. The Spencer–Kodaira map

Let \pi\colon V\to B be a mixed manifold (resp. a \mathbb{C}-analytic mixed space), b\in B, and V_0=\pi^{-1}(b_0). Let T_0 be the tangent space to B at b_0 (resp. the Zariski tangent space). We introduce the following sheaves on V_0:

• \Theta_0: the sheaf of germs of vertical holomorphic fields on V_0 ;
• \Pi_0: the sheaf of germs of locally projectable holomorphic fields on V_0 ; and
• \Lambda_0: the sheaf \pi^*T_0, i.e. the sheaf of germs of locally constant maps from V_0 to T_0.

We have an exact sequence of sheaves on V_0 0 \to \Theta_0 \to \Pi_0 \to \Lambda_0 \to 0 that gives rise to the long exact sequence in cohomology \ldots \to \mathrm{H}^0(V_0;\Pi_0) \to \mathrm{H}^0(V_0;\Lambda_0) \xrightarrow{\delta} \mathrm{H}^1(V_0;\Theta_0) \to \ldots. We also have a canonical map \iota\colon T_0 \to \mathrm{H}^0(V_0;\Lambda_0) that is injective if V_0 is non-empty, and surjective if V_0 is connected.

The Spencer–Kodaira map is the composition \rho_0 = \delta\circ\iota\colon T_0 \to \mathrm{H}^1(V_0;\Theta_0).

This map is an essential tool in the local study of deformations of \mathbb{C}-analytic varieties. Note that \Theta_0 is exactly the sheaf of germs of holomorphic fields of tangent vectors to V_0, and thus depends only on V_0, while T_0 depends only on the base. Also, \Theta_0 is a coherent analytic sheaf on V_0, and, if V_0 is compact, then \mathrm{H}^1(V_0;\Theta_0) is a finite-dimensional vector space over \mathbb{C} [1]. We thus see that, in this case (which is the only case where we can say anything non-trivial), \rho_0 might be possible to calculate.

It is clear that, if the given mixed manifold is trivial (i.e. if V=B\times V_0, with \pi being the projection to B), then the map \rho_0 is zero. The next talk aims to show that, in a certain sense, \rho indicates the non-triviality of V in a neighbourhood of V_0.

# I. The map \widetilde{\rho}

All throughout this talk, B is a \mathscr{C}^\infty manifold (resp. \mathbb{R}-analytic, resp. \mathbb{C}-analytic); \pi\colon V\to B denotes a proper mixed manifold; b_0 is a point of B; and V_0=\pi^{-1}(b_0) is thus a compact \mathbb{C}-analytic manifold.

Let \widetilde{\Theta} (resp. \widetilde{\Pi}) be the sheaf of germs of vertical holomorphic (resp. locally projectable holomorphic) vector fields on V. The quotient sheaf \widetilde{\Lambda}=\widetilde{\Pi}/\widetilde{\Theta} is exactly the inverse image under \pi of the sheaf \widetilde{T} of germs of \mathscr{C}^\infty fields (resp. …) of tangent vectors on B.

For every open subset U of B, set V_U=\pi^{-1}(U). The exact sequence 0 \to \widetilde{\Theta} \to \widetilde{\Pi} \to \widetilde{\Lambda} \to 0 of sheaves on V_U gives rise to a homomorphism \widetilde{\rho}_U\colon \mathrm{H}^0(U;\widetilde{T}) \xrightarrow{\pi_*} \mathrm{H}^0(V_U;\widetilde{\Lambda}) \xrightarrow{\delta} \mathrm{H}^1(V_U;\widetilde{\Theta}). Let \mathrm{R}^1\pi_*\widetilde{\Theta} be the sheaf on B defined by the presheaf U\mapsto\mathrm{H}^1(V_U;\widetilde{\Theta}). Then \widetilde{\rho} becomes a homomorphism of sheaves on B: \widetilde{\rho}\colon \widetilde{T} \to \mathrm{R}^1\pi_*\widetilde{\Theta}. In particular, we have a homomorphism \widetilde{\rho}_0\colon \widetilde{T}_0 \to \mathrm{R}^1\pi_*\widetilde{\Theta} = \mathrm{H}^1(V_0;\widetilde{\Theta}) where \widetilde{T}_0 is the vector space of germs at b_0 of fields of tangent vectors to B. Finally, we have a commutative diagram \begin{CD} \widetilde{T}_0 @>{\widetilde{\rho}_0}>> \mathrm{H}^1(V_0;\widetilde{\Theta}) \\@V{\varepsilon}VV @VV{\varepsilon}V \\T_0 @>>{\rho_0}> \mathrm{H}^1(V_0;\Theta_0) \end{CD} where \rho_0 is the Spencer–Kodaira map .

For the proper mixed manifold \pi\colon V\to B to be locally trivial in a neighbourhood of the point b_0\in B, it is necessary and sufficient for the map \widetilde{\rho}_0\colon\widetilde{T}_0\to\mathrm{H}^1(V_0;\widetilde{\Theta}) to be zero.

Proof.

1. (Necessity). If \pi\colon V\to B is locally trivial at b_0, then, for every open subset U of B over which V is trivial, we have \widetilde{\Pi}=\widetilde{\Lambda}\oplus\widetilde{\Theta} on V_U, and so \delta\colon\mathrm{H}^0(V_U;\widetilde{\Lambda})\to\mathrm{H}^0(V_U;\widetilde{\Theta}) is zero.

2. (Sufficiency). Let (\eta_1,\ldots,\eta_p) be \mathscr{C}^\infty vector fields (resp. …) on a neighbourhood of b_0 in B, such that (\eta_1(b_0),\ldots,\eta_p(b_0)) forms a basis of the tangent space T_0 to B at b_0. It then follows from the hypothesis that the map \mathrm{H}^0(V_0;\widetilde{\Pi}) \to \mathrm{H}^0(V_0;\widetilde{\Lambda}) is surjective. So let (\xi_1,\ldots,\xi_p) be projectable holomorphic vector fields on a neighbourhood of V_0 in V, that project to (\eta_1,\ldots,\eta_p). Let f be the map defined on a neighbourhood of \{0\}\times V_0 in \mathbb{R}^p\times V_0 (resp. \mathbb{C}^p\times V_0) by f(t_1,\ldots,t_p,y) = e^{\xi_1}(t_1,e^{\xi_2}(\ldots,e^{\xi_p}(t_p,y)\ldots)). It follows from the proposition stated in [2, Section III.2] that f induces an isomorphism of mixed manifolds from U\times V_0 to \pi^{-1}(f_1(U)) over f_1, where U is a sufficiently small cubical neighbourhood of 0 in \mathbb{R}^p, and f_1 is the map from U to B defined by f_1(t_1,\ldots,t_p) = e^{\eta_1}(t_1,e^{\eta_2}(\ldots,e^{\eta_p}(t_p,b_0)\ldots)), which proves the theorem.

# II. The regular case

For all b\in B, set V_b=\pi^{-1}(b). Consider the family \{\mathrm{H}^1(V_b;\Theta_b)\}_{b\in B} of finite-dimensional \mathbb{C}-vector spaces, and, for all b\in B, the map \varepsilon_b\colon \mathrm{H}^1(V_b;\widetilde{\Theta}) \to \mathrm{H}^1(V_b;\Theta_b).

For every open subset U\subset B, we have a map \widetilde{\varepsilon}_U\colon \mathrm{H}^1(V_U;\widetilde{\Theta}) \to \prod_{b\in U}\mathrm{H}^1(V_b;\Theta_B) that defines, by varying U, a homomorphism from the sheaf \mathrm{R}^1\pi_*\widetilde{\Theta} to the sheaf \Phi on B defined by \Phi(U)=\prod_{b\in U}\mathrm{H}^1(V_b;\Theta_b).

We say that the proper mixed manifold \pi\colon V\to B is regular if

1. the dimension of \mathrm{H}^1(V_b;\Theta_b) does not depend on the point b\in B; and
2. we can endow E=\bigcup_{b\in B}\mathrm{H}^1(V_b;\Theta_b) with the structure of a \mathscr{C}^\infty vector bundle (resp. …) such that \widetilde{\varepsilon} is an isomorphism from the sheaf \mathrm{R}^1\pi_*\widetilde{\Theta} to the sheaf of germs of \mathscr{C}^\infty sections (resp. …) of the bundle E.

In fact, Kodaira and Spencer have shown [7] that, by identifying the \mathrm{H}^1 spaces with spaces of harmonic forms, condition (2) is a consequence of condition (1).

Then Theorem 1 has the following corollary:

For the proper mixed manifold \pi\colon V\to B to be locally trivial, it is necessary and sufficient for it to be regular and, for all b\in B, for the Spencer–Kodaira map \rho_b\colon T_b \to \mathrm{H}^1(V_b;\Theta_b) to be zero.

Indeed, since \widetilde{\varepsilon} is injective, this condition implies that the map \widetilde{\rho}_b\colon \widetilde{T}_b \to \mathrm{H}^1(V_b;\widetilde{\Theta}) is zero for all b.

At the end of this talk, we will construct a counter-example which shows that it is necessary to assume that the mixed manifold is regular.

# III. An example of non-regular deformation: Hopf manifolds

## 1. Hopf manifolds

Let n\geqslant 2 be an integer, and let b be an (n\times n) matrix with coefficients in \mathbb{C}, whose eigenvalues are all of modulus >1. The free group L(b) generated by b acts freely on \widetilde{V}=\mathbb{C}^n\setminus\{0\}, and the quotient space \widetilde{V}/L(b), which we call the Hopf manifold defined by b, is a compact \mathbb{C}-analytic manifold that is homeomorphic to S^{2n-1}\times S^1.

Note that V_b and V_{b'} are isomorphic if and only if there exists some a such that b'=aba^{-1} or b'=ab^{-1}a^{-1} (cf. Appendix).

Let \Theta be the sheaf of germs of holomorphic fields of tangent vectors on V_b.

We can identify \mathrm{H}^0(V_b;\Theta) with the vector space of matrices that commute with b, and \mathrm{H}^1(V_b;\Theta) has the same dimension as this vector space.

Proof. If X is a vector field on an open subset U\subset\widetilde{V}, then b_*(X) is the vector field on the open subset b(U) given by transporting via b, i.e. b_*X(u)=bX(b^{-1}u). Let \mathscr{U}=\{U_i\} be a cover of V by simply connected Stein open subsets; for all i, set \widetilde{U}_i=\chi^{-1}\{U_i\}, where \chi is the canonical map from \widetilde{V} to V_b. The cover \widetilde{\mathscr{U}}=\{\widetilde{U}_i\} of \widetilde{V} consists of Stein open subsets that are invariant under b (not necessarily connected, but this doesn’t matter). Then b_* defines a map, again denoted by b_*, from the group of cochains C^\bullet(\widetilde{V},\widetilde{U};\Theta) to itself.

We have the exact sequence 0 \to C^\bullet(V_b,\mathscr{U};\Theta) \xrightarrow{\chi^*} C^\bullet(\widetilde{V},\widetilde{U};\Theta) \xrightarrow{1-b_*} C^\bullet(\widetilde{V},\widetilde{U};\Theta) \to 0.

Proof. The only thing that we need to verify is that the map 1-b_* is surjective. For all (i_0,\ldots,i_q), let U'_{i_0,\ldots,i_q} be an open subset of \widetilde{V} such that \chi\colon U'_{i_0,\ldots,i_q} \to U_{i_0,\ldots,i_q} is a homeomorphism. The \widetilde{U}_{i_0,\ldots,i_q} is a disjoint union of the b_*^p U'_{i_0,\ldots,i_q}, where p\in\mathbb{Z}, and every \gamma\in C^q(\widetilde{V},\widetilde{U};\Theta) can be written in the form \gamma=\gamma_1-\gamma_2, with \gamma_1=0 on b^p(U'_{i_0,\ldots,i_q}) for p<0, and \gamma_2=0 for p\geqslant 0. Set \beta = \sum_{p\geqslant 0} b_*^p\gamma_1 + \sum_{p<0} b_*^p\gamma_2 (which is a locally finite sum). Then \beta-b_*\beta=\gamma, whence Lemma 1.

Now, to finish the proof of Proposition 2. From Lemma 1, we have the following exact sequence: 0 \to \mathrm{H}^0(V_b;\Theta) \xrightarrow{\chi^*} \mathrm{H}^0(\widetilde{V};\Theta) \xrightarrow{1-b_*} \mathrm{H}^0(\widetilde{V};\Theta) \xrightarrow{\delta_*} \mathrm{H}^1(V_b;\Theta) \xrightarrow{\chi^*} \mathrm{H}^1(\widetilde{V}y\Theta) \xrightarrow{1-b_*} \mathrm{H}^1(\widetilde{V};\Theta). We can show that \chi^*\colon \mathrm{H}^1(V_b;\Theta) \to \mathrm{H}^1(\widetilde{V};\Theta) is zero: if n>2, it is evident, since \mathrm{H}^1(\widetilde{V};\Theta)=0; if n=2, then a direct calculation on the cochains of a cover of \widetilde{V} by two Stein open subsets shows that 1-b_*\colon \mathrm{H}^1(\widetilde{V};\Theta) \to \mathrm{H}^1(\widetilde{V};\Theta) is bijective.

Now \mathrm{H}^0(\widetilde{V};\Theta) is the space of holomorphic vector fields on \widetilde{V}, but such a field extends to a holomorphic vector field on \mathbb{C}^n, and \mathrm{H}^0(\widetilde{V},\Theta)=L\oplus M, where L is the space of fields of linear vectors, and M is the space of fields of second-order vectors at 0. The subspaces L and M are invariant under b_*, and 1-b_*\colon M\to M is an isomorphism. Then Proposition 2 follows from remarking that, if an element of L is represented by a matrix a, then b_*a=bab^{-1}.

## 2. Mixed manifolds whose fibres are Hopf manifolds

Let B be the set of all (n\times n) matrices with coefficients in \mathbb{C} with eigenvalues all of modulus >1. This is an open subset of \mathbb{C}^{n^2}. Let \alpha be the transformation from B\times\widetilde{V} to itself defined by \alpha(b,x)=(b,b(x)). The free group L(\alpha) generated by \alpha acts linearly on B\times\widetilde{V}, and the quotient V=B\times\widetilde{V}/L(\alpha) is a \mathbb{C}-analytic manifold. By endowing it with the projection \pi\colon V\to B induced by the projection \pi_1\colon B\times\widetilde{V}\to B after passing to the quotient, we obtain a \mathbb{C}-analytic mixed manifold that is proper, but not regular. Indeed, condition 1 of the definition of regular mixed manifolds is not satisfied: for example, for n=2, the dimension of \mathrm{H}^1(V_b;\Theta) is 4 if b is a scalar matrix, but 2 in all other cases.

Note that the dimension of \mathrm{H}^1(V_b;\Theta_b) is an upper semi-continuous function of b, and that the set of b such that \dim\mathrm{H}^1(V_b;\Theta_b)\geqslant k is a closed analytic subspace of B. This is a general result, that we hope to be able to prove in a later talk of this seminar.

## 3. Calculation of \rho

We have T_b=\operatorname{Hom}(\mathbb{C}^n,\mathbb{C}^n)=L\subset\mathrm{H}^0(\widetilde{V};\Theta), and we defined, to prove Proposition 2, a surjective map \delta_*\colon L\to\mathrm{H}^1(V_b;\Theta).

The Spencer–Kodaira map \rho is given, for the mixed manifold studied in this section, by \rho(a) = \delta_*(ab^{-1}). In particular, it is surjective, and its kernel is the space of matrices of the form [\ell,b] for \ell\in L.

Proof. Let a\in T_b=L. Let \{U_i\} be a cover of V_b by simply connected Stein open subsets, and, for each i, let U'_i be a connected component of \widetilde{U}_i.

Let \eta'_i be the projectable holomorphic field on U'_i defined by \eta'_i(x)=(a,0); let \widetilde{\eta}_i be the projectable holomorphic field on \widetilde{U}_i defined by \widetilde{\eta}_i=\alpha_*^k\eta'_i on b^k(U'_i); and let \eta_i be the projectable holomorphic field on U_i corresponding to \widetilde{\eta}_i. By definition, \rho(a) is the cohomology class of the cochain \{\theta_{ij}\}, where \theta_{ij}=\eta_j-\eta_i is a vertical holomorphic field on U_{ij}.

Set \widetilde{\eta}_i(x)=(a,\beta_i(x)). Then \beta\in C^0(\widetilde{V};\Theta), and we have (1-b_*)\beta=ab^{-1}\in L\subset\mathrm{H}^0(\widetilde{V};\Theta). Indeed, \alpha_*\eta=\eta, \alpha_*\eta_i(b_{-1}x)=\eta_i(x), and \alpha_*(a,\beta(b^{-1}x)) = (a,\beta(x)), whence ab^{-1}x + b\cdot\beta(b^{-1}x) = \beta(x). We thus deduce that \theta=\delta_*(ab^{-1}), which proves Proposition 3.

## 4. A counter-example

Take n=2, and \sigma\in\mathbb{C} such that |\sigma|>1. Let B'\subset B be the set of matrices of the form \begin{pmatrix} \sigma & t \\0 & \sigma \end{pmatrix} where t\in\mathbb{C}, and let V'=\pi^{-1}(B') be the mixed manifold induced by V over V'; now B' is a line, and its tangent space T'_b at b is generated, for all b, by a=\begin{pmatrix}0&1\\0&0\end{pmatrix}. It follows from Proposition 3 that the Spencer–Kodaira map \rho'\colon T_b(B') \to \mathrm{H}^1(V_b;\Theta) is zero if and only if b \neq b_0 = \begin{pmatrix} \sigma & 0 \\0 & \sigma \end{pmatrix} since, if b\neq b_0, then a=[\ell,b], where \ell=\begin{pmatrix}t^{-1}&0\\0&0\end{pmatrix}; and if b=b_0, then \rho' is injective.

We can also see that V' is trivial on B'\setminus\{b_0\}.

Let \varphi\colon\mathbb{C}\to B'\subset B be the map defined by \varphi(t) = \begin{pmatrix} \sigma & t^2 \\0 & \sigma \end{pmatrix} and let V^\varphi be the mixed manifold given by the inverse image of V under \varphi. The Spencer–Kodaira map \rho_t^\varphi from \mathbb{C} to \mathrm{H}^1(V_{\varphi(t)};\Theta) is the composition \rho'_{\varphi(t)}\circ\mathrm{D}\varphi\colon \mathbb{C} \to T'_{\varphi(t)} \to \mathrm{H}^1(V_{\varphi(t)};\Theta), and this is zero for all t, since, if t\neq0, then \rho'_{\varphi(t)} is zero; and, if t=0, then \mathrm{D}\varphi is zero.

However, the mixed manifold V^\varphi is not locally trivial, since V_0^\varphi is not isomorphic to V_t^\varphi for t\neq0.

## 5. Question (K. Srinivasacharyulu)

We know that the Hopf manifolds are non-K"{a}hler, and thus non-algebraic. For n=2, the manifold V_b admits non-constant meromorphic functions if and only if b can be diagonalised with eigenvalues \sigma_1 and \sigma_2 satisfying \sigma_1^p=\sigma_2^q for some integers p and q (and there is then the function x_1^px_2^{-q}). The set of b satisfying this property is neither open nor closed, but it is a countable union of closed analytic subspaces. An analogous phenomenon arises for deformations of complex tori. Is this result general?

# Appendix

With the notation of §III.1, let f\colon V_b\to V_{b'} be an isomorphism of \mathbb{C}-analytic manifolds. This lifts to an isomorphism of universal coverings \widetilde{f}\colon \mathbb{C}^n\setminus\{0\} \to \mathbb{C}^n\setminus\{0\}. By Hartog, \widetilde{f} extends to an isomorphism g\colon\mathbb{C}^n\to\mathbb{C}^n. We necessarily have g(bz) = (b')^kg(z) \tag{$*$} where z\in\mathbb{C}^n, and k is an integer; the same property, applied to the inverse map of g, shows that k=\pm1. Let a be the linear map that is tangent to g at the origin; the identity (*) then gives \begin{aligned} ab &= (b')^ka \\k &= \pm1 \end{aligned} whence b' = aba^{-1} \quad\text{or}\quad b'= ab^{-1}a^{-1}.

# Introduction

Let V_0 be a compact complex-analytic manifold, and let \Theta be the sheaf of germs of holomorphic fields of tangent vectors. We ask the following question: given an element a\in\mathrm{H}^1(V_0,\Theta), does there exists a deformation of V_0, with a non-singular base (i.e. a fibred mixed manifold \pi\colon V\to B, with b_0\in B, along with an isomorphism V_0\xrightarrow{\cong}\pi^{-1}(b_0)), such that a is the image, under the map \rho defined in [Talk no. 2], of a vector v that is tangent to B at b_0? An element a\in\mathrm{H}^1(V_0,\Theta) for which the answer is positive is called a deformation vector. We will give a necessary condition for a to be a deformation vector; this condition is written [a\smile a]=0. We will then give an example where this condition is not satisfied.

# I. Exact sequences of sheaves of algebras

Let K be a commutative ring, and let \Phi, \Phi_1, and \Phi_2 be sheaves of K-modules on some space X, and suppose that we have some given homomorphism \Phi_1\otimes\Phi_2\to\Phi, written as a product. We define, for any cover {\mathscr{U}} of X, the cup product \smile\colon C^p(X,{\mathscr{U}};\Phi_1) \otimes C^q(X,{\mathscr{U}};\Phi_2) \to C^{p+q}(X,{\mathscr{U}};\Phi) by the formula (\alpha\smile\beta)_{i_0,\ldots,i_{p+q}} = \alpha_{i_0,\ldots,i_p}\cdot\beta_{i_p,\ldots,i_{p+q}}. We have the relation \mathrm{d}(\alpha\smile\beta) = \mathrm{d}\alpha\smile\beta + (-1)^p\alpha\smile\mathrm{d}\beta. This induces a cup product on the cohomology of the cover {\mathscr{U}}, and, by passing to the inductive limit over open covers, a cup product \smile\colon \mathrm{H}^p(X;\Phi_1) \otimes \mathrm{H}^q(X;\Phi_2) \to \mathrm{H}^{p+q}(X;\Phi).

A sheaf of algebras on X is a sheaf of modules \Phi on X endowed with a product \Phi\otimes\Phi\to\Phi (which we do not assume to be either commutative nor associative).

If f\colon\Phi\to\Psi is a homomorphism of sheaves of algebras, then the kernel \Phi' of f is a sheaf of two-sided ideals of \Phi, i.e. we have products \Phi'\otimes\Phi\to\Phi' and \Phi\otimes\Phi'\to\Phi' such that the two diagrams \begin{CD} \Phi'\otimes\Phi @>>> \Phi' \\@VVV @VVV \\\Phi\otimes\Phi @>>> \Phi \end{CD} \qquad \begin{CD} \Phi\otimes\Phi' @>>> \Phi' \\@VVV @VVV \\\Phi\otimes\Phi @>>> \Phi \end{CD} both commute.

Let 0\to\Phi'\to\Phi\to\Phi''\to0 be an exact sequence of sheaves of algebras on X; let a\in\mathrm{H}^p(X;\Phi''). Then \delta a\in\mathrm{H}^{p+1}(X;\Phi'), and, for any class b\in\mathrm{H}^q(X;\Phi'), we have \delta a\smile b=0.

Proof. Let {\mathscr{U}} be a cover of X such that a and b are represented by cocycles \alpha and \beta (respectively), and such that \alpha lifts to a cochain \eta\in C^p(X,{\mathscr{U}};\Phi). Then \delta\eta is a cocycle in C^{p+1}(X,{\mathscr{U}};\Phi') whose class in \mathrm{H}^{p+1}(X;\Phi') is, by definition, \delta a, and \delta a\smile b is the class of \delta\eta\smile\beta. But \delta(\eta\smile\beta)=\delta\eta\smile\beta, and \eta\smile\beta is a cochain in C^{p+q}(X,{\mathscr{U}};\Phi'), since \Phi' is a sheaf of ideals. So the cocycle \delta\eta\smile\beta is cohomologous to 0 in \mathrm{H}^{p+q+1}(X;\Phi'), which proves the proposition.

# II. The primary obstruction

Let V_0 be a complex-analytic manifold, and \Theta_0 the sheaf of germs of holomorphic fields of tangent vectors. Then \Theta_0 is a sheaf of Lie algebras, and, if a,b\in\mathrm{H}^\bullet(V_0,\Theta_0), then we denote by [a\smile b] the cup product defined by the bracket [-,-]\colon\Theta_0\otimes\Theta_0\to\Theta_0. It satisfies [b\smile a] = (-1)^{pq+1}[a\smile b] for a\in\mathrm{H}^p(V_0,\Theta_0) and b\in\mathrm{H}^q(V_0,\Theta_0).

Let \pi\colon V\to B be a mixed manifold, b_0 a point of B, V_0=\pi^{-1}(b_0), and let \rho_0\colon T_0\to\mathrm{H}^1(V_0,\Theta_0) be Spencer–Kodaira map. Then, if u and v are tangent vectors of B at b_0, we have [\rho_0(u)\smile\rho_0(v)] = 0.

Let V_0 be a complex-analytic manifold, and \Theta the sheaf of germs of holomorphic fields of tangent vectors of V_0. If a\in\mathrm{H}^1(V_0,\Theta) is a deformation vector, then [a\smile a]=0.

Proof. (Proof of the Corollary). This is simply a particular case of Theorem 1; note that [a\smile b] is a symmetric bilinear map from \mathrm{H}^1\otimes\mathrm{H}^1 to \mathrm{H}^2, and that we are in characteristic 0\neq2.

Proof. (Proof of Theorem 1). Consider the following sheaves on V_0:

• \Theta_0: the sheaf of germs of vertical holomorphic fields on V_0;
• \widetilde{\Theta}_0: the sheaf of germs of vertical holomorphic fields on V;
• \Pi_0: the sheaf of germs of locally projectable holomorphic fields on V_0;
• \widetilde{\Pi}_0: the sheaf of germs of locally projectable holomorphic fields on V;
• \Lambda_0: the sheaf \pi^*T_0, where T_0 is the tangent space of B at b_0; and
• \widetilde{\Lambda}_0: the sheaf \pi^*\widetilde{T}_0, where \widetilde{T}_0 is the space of germs at b_0 of fields on B of tangent vectors of B.

We have the following diagram: \begin{CD} 0 @>>> \widetilde{\Theta}_0 @>>> \widetilde{\Pi}_0 @>>> \widetilde{\Lambda}_0 @>>> 0 \\@. @V{\varepsilon}VV @V{\varepsilon}VV @V{\varepsilon}VV @. \\0 @>>> \Theta_0 @>>> \Pi_0 @>>> \Lambda_0 @>>> 0 \end{CD} whence we obtain the following commutative diagram: \begin{CD} \widetilde{T}_0 @>{\widetilde{\rho}}>> \mathrm{H}^1(V_0;\widetilde{\Theta}) \\@V{\varepsilon}VV @VV{\varepsilon}V \\T_0 @>>{\rho}> \mathrm{H}^1(V_0;\Theta_0) \end{CD}

Let u,v\in T_0 be fixed tangent vectors of B at b_0. We can always find vector fields \widetilde{u} and \widetilde{v} on B that take the values u and v (respectively) at b_0; \epsilon(\widetilde{u})=u and \epsilon(\widetilde{v})=v. The exact sequence 0 \to \widetilde{\Theta}_0 \to \widetilde{\Pi}_0 \to \widetilde{\Lambda}_0 \to 0 is a sequence of homomorphisms of sheaves of Lie algebras, and so [\widetilde{\rho}(\widetilde{u})\smile\widetilde{\rho}(\widetilde{v})] = 0 by Proposition 1. But \epsilon\colon\widetilde{\Theta}_0\to\Theta_0 is also a homomorphism of sheaves of Lie algebras, and the diagram \begin{CD} \mathrm{H}^1(V_0,\widetilde{\Theta}_0)\otimes\mathrm{H}^1(V_0,\widetilde{\Theta}_0) @>{[-\smile-]}>> \mathrm{H}^2(V_0,\widetilde{\Theta}_0) \\@V{\varepsilon\otimes\varepsilon}VV @VV{\varepsilon}V \\\mathrm{H}^1(V_0,\Theta_0)\otimes\mathrm{H}^1(V_0,\widetilde{\Theta}_0) @>>{[-\smile-]}> \mathrm{H}^2(V_0,\Theta_0) \end{CD} commutes. We thus deduce that [\rho(u)\smile\rho(v)]=0.

1. We make essential use of the fact that \epsilon\colon\widetilde{T}_0\to T_0 is surjective, and thus of the fact that B has no singularities.
2. We actually have [\rho(u)\smile b]=0 for all u\in T_0, for any class b\in\mathrm{H}^1(V_0,\Theta_0) that is in the image of \mathrm{H}^1(V_0,\widetilde{\Theta}_0) under \epsilon. In particular, for an element a\in\mathrm{H}^1(V_0,\Theta_0) to be a regular deformation vector (in the sense of [Talk no. 3]), it is necessary and sufficient for [a\smile b]=0 for all b\in\mathrm{H}^1(V_0,\Theta_0).

If V_0 is a compact complex-analytic manifold, and a\in\mathrm{H}^1(V_0,\Theta), then we call [a\smile a]\in\mathrm{H}^2(V_0,\Theta) the primary obstruction to the deformation of V_0 along a. For a to be a deformation vector, it is necessary that this primary obstruction be zero; but it is not sufficient: we can define a sequence of set-theoretic maps \omega_n, called obstructions, with \omega_1\colon\mathrm{H}^1(V_0,\Theta)\to\mathrm{H}^2(V_0,\Theta) given by \omega_1(a)=[a\smile a], and with \omega_{k+1} defined on the subset of \mathrm{H}^1(V_0,\Theta) where \omega_k vanishes, with values in varying quotients2 of \mathrm{H}^2(V_0,\Theta), and a necessary condition for a to be a deformation vector is that all the \omega_k(a) be defined and real. I do not know if this condition is sufficient. Kodaira, Spencer, and Nijenhuis [5] have shown that, if \mathrm{H}^2(V_0,\Theta)=0, then every element of \mathrm{H}^1(V_0,\Theta) is a deformation vector. In this case, we even have a locally universal deformation whose base is a manifold, and \rho is an isomorphism from the tangent space of this manifold to \mathrm{H}^1(V_0,\Theta)

# III. An example of obstruction

## 1. The manifold V_0

Let X=E/\Gamma be a 2-dimensional complex torus, i.e. E\cong\mathbb{C}^2 and \Gamma\cong\mathbb{Z}^4, and let D the be projective line \mathbb{P}^1\mathbb{C}. Set V_0=X\times D. The sheaf \Theta of holomorphic fields of tangent vectors of V_0 is the direct sum of the sheaves of Lie algebras \Theta_1 and \Theta_2, where \begin{aligned} \Theta_1 &= {\mathcal{O}}\otimes_{{\mathcal{O}}_X}\pi_1^*\Theta_X \\\Theta_2 &= {\mathcal{O}}\otimes_{{\mathcal{O}}_D}\pi_2^*\Theta_D \end{aligned} where \pi_1\colon V_0\to X and \pi_2\colon V_0\to D are the projections, {\mathcal{O}}, {\mathcal{O}}_X, and {\mathscr{D}} are the structure sheaves (sheaves of local rings), and \Theta_X and \Theta_D are the sheaves of germs of holomorphic fields of tangent vectors of X and D (respectively). We are mostly interested in \Theta_2. Also, \mathrm{H}^1(V_0,\Theta_2) is given by the Künneth exact sequence: 0 \to \mathrm{H}^0(X,{\mathcal{O}}_X)\otimes\mathrm{H}^1(D,\Theta_D) \to \mathrm{H}^1(V_0,\Theta_2) \to \mathrm{H}^1(X,{\mathcal{O}}_X)\otimes\mathrm{H}^0(D,\Theta_D) \to 0. But we know that \mathrm{H}^0(D,\Theta_D) is the Lie algebra {\mathfrak{a}} of the group A = \operatorname{GL}(2,\mathbb{C})/\mathbb{C}^* = \operatorname{SL}(2,\mathbb{C})/\{\pm1\} of automorphisms of D, and that \mathrm{H}^1(D,\Theta_D)=0, as we can easily see by taking a cover of D by two open subsets. We have already seen (in [Talk no. 1]) that, if X=E/\Gamma, then \mathrm{H}^1(X,{\mathcal{O}})=\operatorname{Hom}(\Gamma,\mathbb{C})/\operatorname{Hom}_{\mathbb{C}}(E,\mathbb{C}) is of dimension 2. So \mathrm{H}^1(V_0,\Theta_2)=\mathrm{H}^1(X,{\mathcal{O}})\otimes{\mathfrak{a}} is of dimension 6. The cup product \mathrm{H}^1(V_0,\Theta_2)\otimes\mathrm{H}^1(V_0,\Theta_2) \to \mathrm{H}^2(V_0,\Theta_2) is given by the formula [(\gamma\otimes\alpha)\smile(\gamma'\otimes\alpha')] = (\gamma\smile\gamma')\otimes[\alpha,\alpha']. The cone of elements \varphi\in\mathrm{H}^1(V_0,\Theta_2) such that [\varphi\smile\varphi]=0 can be identified with the cone of rank 1 tensors in \mathrm{H}^1(X,{\mathcal{O}})\otimes{\mathfrak{a}}. Indeed, if \varphi=\gamma\otimes\alpha, then [\varphi\smile\varphi] = (\gamma\smile\gamma)\otimes[\alpha,\alpha] = 0\otimes0 = 0 and, if \varphi is not a simple tensor, then we have \varphi = \gamma\otimes\alpha + \gamma'\otimes\alpha' with \gamma and \gamma' independent, and \alpha and \alpha' independent, so [\varphi\smile] = 2(\gamma\smile\gamma')\otimes[\alpha,\alpha'] \neq 0.

## 2. The mixed space V

In this example, every element of \mathrm{H}^1(V_0,\Theta_2) whose primary obstruction is zero is a deformation vector. More precisely:

There exists a mixed space \pi\colon V\to B and a point b_0\in B such that

1. \pi^{-1}(b_0)=V_0 (the manifold defined in §III.1);
2. there exists an isomorphism \sigma from a \mathbb{C}-analytic space B to the cone of elements \varphi\in\mathrm{H}^1(V_0,\Theta_2) such that [\varphi\smile\varphi]=0; and
3. for every subspace B' of B that has no singularities at b_0, the Spencer–Kodaira map \rho from the tangent space of B' at b_0 to \mathrm{H}^1(V_0,\Theta) agrees with \sigma\colon B'\to\mathrm{H}^1(V_0,\Theta_2).

Let H be the analytic space of homomorphisms from \Gamma to {\mathfrak{a}} whose images are contained in a vector subspace of {\mathfrak{a}} that is 1-dimensional over \mathbb{C} (i.e. (4\times2) matrices of rank 1 with coefficients in \mathbb{C}). For every h\in H, e\circ h is a homomorphism from \Gamma to A, where e\colon{\mathfrak{a}}\to A denotes the exponential map, and we construct a manifold V_h that is fibred over X with fibre D as follows: V_h is the quotient of E\times D by the equivalence relation defined by \Gamma acting via \gamma\star(x,y) = (x+\gamma,((e\circ h)(\gamma))\cdot y). These manifolds are the fibres of a mixed space W\to H, where W is the quotient of H\times E\times D by the equivalence relation defined by \Gamma acting via \gamma\star(h,x,y) = (h,x+y,(e\circ h(y))\cdot y). We now place the following equivalence relation on H: we have h'\sim h if and only if (h'-h) extends to an \mathbb{C}-linear map f\colon E\to{\mathfrak{a}}. Note that, if h'(\Gamma) and h(\Gamma) are contained in the same subspace L of {\mathfrak{a}} of dimension 1 over \mathbb{C} (or if h'\sim h), then we also have f(E)\subset L (or h\sim0 and h'\sim0). In both cases, V_h and V_{h'} are isomorphic, and we have an isomorphism i_{h',h}\colon V_h\to V_{h'} defined by i_{h',h}(x,y) = (x,e\circ f(x)\cdot y) (in the first case), or i_{h',h} = i_{h',0}\circ i_{0,h} (in the second case). If h, h', and h'' are in the same class, then we have i_{h''h}=i_{h''h'}\circ i_{h'h}, and we can place on W the equivalence relation (h',z') \sim (h,z) \iff h'\sim h\text{ or }z'=i_{h'h}z for h,h'\in H, z\in V_h, and z'\in V_{h'}.

Let B and V be the quotients of H and W (respectively) by these equivalence relations. We have a projection V\to B. To show that the structures of a \mathbb{C}-analytic space on H and W induce structures of a \mathbb{C}-analytic space on their quotients B and V, it suffices to remark that we can lift B to a analytic subspace of H: let, for example, (\gamma_1,\gamma_2,\gamma_3,\gamma_4) be a basis of \Gamma such that (\gamma_1,\gamma_2) is a basis of E over \mathbb{C}; then each class b\in B contains exactly one element h\in H such that h(\gamma_1) = h(\gamma_2) = 0.

## 3. Calculating \rho_0

Let T be the Zariski tangent space of B at b_0, i.e. the dual of {\mathfrak{I}}/{\mathfrak{I}}^2, where {\mathfrak{I}} is the ideal of germs at b_0 of analytic functions on B that are zero at b_0. Then T_0 can be identified with \operatorname{Hom}(\Gamma,a)/\operatorname{Hom}_{\mathbb{C}}(E,a). Also, \begin{aligned} \mathrm{H}^1(V_0,\Theta) &= \mathrm{H}^1(V_0;\Theta_1) \oplus \mathrm{H}^1(V_0;\Theta_2) \\&= \big(\mathrm{H}^1(X;{\mathcal{O}}) \otimes E\big) \oplus \big(\mathrm{H}^1(X;{\mathcal{O}})\otimes a\big), \end{aligned} and the second term of this term can be identified with the quotient \operatorname{Hom}(\Gamma,a)/\operatorname{Hom}_{\mathbb{C}}(E,a). We are going to show that the map \rho_0\colon T_0\to\mathrm{H}^1(V_0;\Theta) is exactly the canonical injection defined by these identifications.

Let u\in T_0=\operatorname{Hom}(\Gamma,\alpha)/\operatorname{Hom}(E,\alpha) be the class of an element h\in\operatorname{Hom}(\Gamma,\alpha), which we suppose to be of rank 1. Then we can write h in the form \eta\otimes\sigma, where \eta\in\operatorname{Hom}(\Gamma,\mathbb{C}), \sigma\in\alpha, and we can consider h as a tangent vector to H at 0. Let \overline{h} be the field of tangent vectors to H\times E\times D at 0\times E\times D that projects onto h, and thus whose components over E\times D are zero. Let (U_i) be a cover of X=E/\Gamma by simply connected open subsets, and choose, for each i, a component \widetilde{U}_i of the inverse image of U_i in E. We will denote by v_i the image over U_i\times D of the field \overline{h}|\widetilde{U}_i\times D. This is a projectable holomorphic field on 0\times U_i\times D of tangent vectors of H\times U_i\times D, and we set w_{ij}=v_j-v_i, so that w_{ij} is a vertical holomorphic field on U_{ij}\times D, and these fields form a cocycle whose cohomology class will be, by definition, \rho_0(u).

Let x\in U_{ij}, and let \widetilde{x}_i and \widetilde{x}_j be its inverse image in \widetilde{U}_i and \widetilde{U}_j (respectively). We have that \widetilde{x}_j=\widetilde{x}_i+\gamma_{ij}(x), where \gamma_{ij}(x)\in\Gamma, and w_{ij}(x) = \overline{h}(\widetilde{x}_j) - [\gamma_{ij}(x)]_*(\overline{h}(\widetilde{x}_i)) = -h(\gamma_{ij}(x)) \in\alpha. Now w_{ij} is a vector field on D, and so (w_{ij}) \in \mathrm{Z}^1(V_0,(U_i\times D);\Theta_2), and w_{ij} is of the form \zeta\otimes\alpha, where \zeta\in\mathrm{Z}^1(V_0,(U_i\times D);{\mathcal{O}}) is the cocycle defined by \zeta_{ij}(x)=-\eta(\gamma_{ij}(x)). This is a cocycle whose cohomology class is (up to a sign) the element of \mathrm{H}^1(V_0,{\mathcal{O}}) that is identified with the class \eta in \operatorname{Hom}(\Gamma,\mathbb{C})/\operatorname{Hom}_{\mathbb{C}}(E,\mathbb{C}). QED.

# I. Definition of obstructions

## 1. The sheaf of germs of vertical automorphisms

Let V_0 be a \mathbb{C}-analytic manifold, which we assume to be compact, and B a \mathbb{C}-analytic space, and let b_0\in B. We are going to define a sheaf \Gamma of non-abelian groups on V_0. For every open subset U of V_0, consider the isomorphisms of analytic varieties \gamma\colon W\to W', where W and W' are open subsets of B\times V_0 that contain \{b_0\}\times U, such that the following conditions are satisfied:

1. \pi_1\gamma=\pi_1 is the projection B\times V_0 to B;
2. \gamma is the identity on \{b_0\}\times U.

Then \Gamma(U) consists of equivalence classes of these isomorphisms, where we identify \gamma_1 with \gamma_2 if they agree on a neighbourhood of \{b_0\}\times U.

It is clear that \Gamma(U) is a group under composition of isomorphisms, and that the \Gamma(U) form a sheaf \Gamma of non-abelian groups.

We can identify \mathrm{H}^1(V_0,\Gamma) with the set of classes of deformation germs of V_0 over (B,b_0).

Recall that a deformation germ of V_0 over (B,b_0) is a deformation of V_0 over a neighbourhood of b_0 in B, and that two such deformations (B',b_0,V',\pi',\iota') and (B'',b_0,V'',\pi'',\iota'') are locally equivalent if there exists a neighbourhood W' of (\pi')^{-1}(b_0) in V', a neighbourhood W'' of (\pi'')^{-1}(b_0) in V'', and an isomorphism \varphi from W' to W'' such that the diagram \begin{CD} V_0 @= V_0 \\@VVV @VVV \\W' @>{\varphi}>> W'' \\@V{\pi'}VV @VV{\pi''}V \\B @= B \end{CD} commutes.

Proof. (Proof of Proposition 1). Let (B',b_0,V,\pi,\iota) be a deformation of V_0\$ over a neighbourhood V' of b_0 in B. Then we can find a cover \{U_i\} of V_0 and a cover \{W_i\} of a neighbourhood of \iota(V_0) in V, along with isomorphisms \{h_i\}, where h_i is an isomorphism from a neighbourhood of \{b_0\}\times U_i in B\times V_0 to W_i that agrees with \iota on \{b_0\}\times U_i, and such that \pi\circ h_i=\pi_1.

Set \gamma_{ij}=h^{-1}_i\circ h_j. We can show that the \gamma_{ij} define an element of \Gamma(U_i\cap U_j), and that \gamma_{ij}\circ\gamma_{jk}=\gamma_{ik}. The \gamma_{ij} thus form a cocycle \gamma\in\operatorname{Z}^1(V_0,\{U_i\};\Gamma). Such a cocycle is said to be associated to the deformation. It will still be associated to the deformation if pass to a finer cover. Let (B',b_0,V',\pi',\iota') be a deformation that is locally equivalent to the first, and let \gamma' be a cocycle associated to this deformation. We can suppose, by refining the covers if necessary, that the cocycles \gamma and \gamma' are defined with respect to the same cover \{U_i\} of V_0. Let f be an isomorphism from a neighbourhood of \iota(V_0) in V to a neighbourhood of \iota'(V_0) in V'. Set f_i=(h'_i)^{-1}\circ f\circ h_i. Then f_i\in\Gamma(U_i), and f_i\circ\gamma_{ij} = \gamma'_{ij}\circ f_j. We thus conclude that the cocycles associated to a deformation form a cohomology class that depends only on the local class of the deformation.

Conversely, suppose we have a locally finite cover \{U_i\} of V_0 and a cocycle \gamma\in\operatorname{Z}^1(V_0,\{U_i\};\Gamma). Then \gamma_{ij} can be represented by an isomorphism from an open W_{ij} of B\times V_0 to another open W_{ji}, with the two open subsets both containing \{b_0\}\times U_{ij}. Pick a refinement \{U'_i\} of the cover \{U_i\}, and take some neighbourhood B'' of b_0 in B small enough such that B''\times U'_{ij}\subset W_{ij} for all (i,j), and such that the equality \gamma_{ij}\circ\gamma_{jk}=\gamma_{ik} holds wherever it is defined in B''\times U'_{ijk}. We thus obtain a deformation V of V_0 on B'' by gluing the B''\times U'_i via the \gamma_{ij}.

Finally, we can show that all the above does indeed define a bijection between the set of local classes of deformations of V_0 over (B,b_0) and \mathrm{H}^1(V_0;\Gamma).

## 2. Higher obstructions

For every open subset U\subset V_0, the group \Gamma(U) is naturally filtered: denote by {\mathscr{F}}_k(U) the group of vertical automorphisms that are tangent to the identity up to order k-1. Then \Gamma becomes a filtered sheaf: \Gamma = {\mathscr{F}}_1 \supset {\mathscr{F}}_2 \supset \ldots \qquad\text{and }\bigcap {\mathscr{F}}_k=\{0\}. Set \begin{aligned} {\mathscr{Q}}_k &= \Gamma/{\mathscr{F}}_{k+1} \\{\mathscr{G}}_k &= {\mathscr{F}}_k/{\mathscr{F}})_{k+1} = \operatorname{Ker}({\mathscr{Q}}_k\to{\mathscr{Q}}_{k-1}). \end{aligned} For all k, {\mathscr{G}}_k is a sheaf of abelian groups, which we will write additively. If B=\mathbb{C} and b_0=0 (we then speak of the deformation in one parameter), for all k, {\mathscr{G}}_k can be identified with the sheaf \Theta of germs of vector fields tangent to V_0. In the general case, {\mathscr{G}}_k = {\mathfrak{m}}^k/{\mathfrak{m}}^{k+1}\otimes\Theta where {\mathfrak{m}} is the maximal ideal of the point b_0 in B.

Now, if a\in{\mathscr{F}}_p and b\in{\mathscr{F}}_q, then the commutator aba^{-1}b^{-1} is in {\mathscr{F}}_{p+q}, and this defines a map {\mathscr{G}}_p\otimes{\mathscr{G}}_q\to{\mathscr{G}}_{p+q} which endows {\mathscr{G}}_\bullet=\bigoplus{\mathscr{G}}_k with the structure of a sheaf of Lie algebras that is isomorphic to the tensor product of \Theta with the graded algebra associated to the maximal ideal {\mathfrak{m}} of b_0 in B filtered by powers.

The exact sequence of non-abelian groups 0 \to {\mathscr{G}}_{k+1} \to {\mathscr{Q}}_{k+1} \to {\mathscr{Q}}_k \to 0 in which {\mathscr{G}}_{k+1} is a subgroup of {\mathscr{Q}}_{k+1} contained in its centre gives rise [3] to an exact sequence of pointed sets \mathrm{H}^1(V_0;{\mathscr{Q}}_{k+1}) \to \mathrm{H}^1(V_0;{\mathscr{Q}}_k) \xrightarrow{\delta_k} \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}) i.e. for an element q\in\mathrm{H}^1(V_0,{\mathscr{Q}}_k) to be in the image of \mathrm{H}^1(V_0;{\mathscr{Q}}_{k+1}), it is necessary and sufficient for \delta_k q=0 in \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}). A necessary condition for q to be in the image of \mathrm{H}^1(V_0;\Gamma)\to\mathrm{H}^1(V_0;{\mathscr{Q}}_k) is thus \delta_k q=0 in \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}).

Let q\in\mathrm{H}^1(V_0;{\mathscr{Q}}_i), and let k\geqslant i. We define an obstruction of order k of the element q to be the direct image in \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}) under \delta_k of the inverse image of q in \mathrm{H}^1(V_0;{\mathscr{Q}}_k). It is thus a subset of \mathrm{H}^2(V_0;{\mathscr{G}}_{k+1}). The obstruction is said to be trivial if the identity element belongs to this subset. Being trivial is a necessary and sufficient condition for q to be in the image of \mathrm{H}^1(V_0;{\mathscr{Q}}_{k+1}), and a necessary condition for q to be in the image of \mathrm{H}^1(V_0;\Gamma).

If q is not in the image of \mathrm{H}^1(V_0,{\mathscr{Q}}_k), then its obstruction of order k is empty, and thus non-trivial.

This definition is used most of all in the case of deformations in one parameter (B=\mathbb{C} and b_0=0), where {\mathscr{G}}_{k+1}=\Theta for all k, and {\mathscr{Q}}_1={\mathscr{G}}_1=\Theta. The successive obstructions of an element a\in\mathrm{H}^1(V_0;\Theta) are thus subsets of \mathrm{H}^2(V_0;\Theta), and for a to be a deformation vector, it must be the case that all of its obstructions are trivial. Indeed, the element of \mathrm{H}^1(V_0;\Theta) that corresponds, under the identifications we have made (\Theta={\mathscr{Q}}_1=\Gamma/{\mathscr{F}}_2, and Proposition 1), to a deformation germ is exactly the image under the Spencer–Kodaira map \rho of the canonical basis vector of the tangent space to \mathbb{C} at 0.

# II. Calculation of obstructions

## 1. Relation to the sheaf \Omega

From now on, we work in the case of deformations in one parameter, i.e. B=\mathbb{C} and b_0=0.

Let \Omega be the sheaf of universal enveloping algebras of the Lie algebras of the sheaf \Theta (i.e. \Omega(U) is the universal enveloping algebra of \Theta(U)).

Then \Omega contains \Theta as a subsheaf, and even as a direct factor (by the Poincaré–Birkhoff–Witt Theorem in characteristic 0). For all k, consider the sheaf of algebras \Omega_k=\Omega[t]/(t^{k+1}). For i\leqslant k, we have a map of sheaves of sets \exp_i\colon \Theta \to \Omega_k defined by \exp_i(\Theta) = \sum_p\frac{1}{M} \Theta^p t^p

(Campbell–Hausdorff). We can identify {\mathscr{Q}}_k with the sheaf of multiplicative subgroups of \Omega_k generated by the images of the \exp_i for i\leqslant k.

The proof of this proposition will not be given here. We denote by \Omega_k^\times the sheaf of multiplicative subgroups of \Omega_k consisting of the elements whose constant terms is 1. The commutative diagram of sheaves of (non-abelian) groups \begin{CD} 0 @>>> \Theta @>>> {\mathscr{Q}}_{k+1} @>>> {\mathscr{Q}}_k @>>> 0 \\@. @VVV @VVV @VVV @. \\0 @>>> \Omega @>>> \Omega_{k+1}^\times @>>> \Omega_k^\times @>>> 0 \end{CD} gives rise to a commutative diagram of sets \begin{CD} \mathrm{H}^1(V_0;{\mathscr{Q}}_k) @>{\delta_k}>> \mathrm{H}^2(V_0;\Theta) \\@VVV @VVV \\\mathrm{H}^1(V_0;\Omega_k^\times) @>>{\delta_k}> \mathrm{H}^2(V_0;\Omega) \end{CD} in which \mathrm{H}^2(V_0;\Theta) is a vector subspace of \mathrm{H}^2(V_0;\Omega).

## 2. Calculation of the primary obstruction

Now let a\in\mathrm{H}^1(V_0;\Theta), and let \alpha=(\alpha_{ij}) be a cocycle of the class a (the choice of the cocycle \alpha does not matter, since every cocycle that is cohomologous to a deformation cocycle is itself a deformation cocycle). The corresponding multiplicative cocycle in \Omega_1^\times is (1+\alpha_{ij}t). This cocycle can be lifted to \Omega_i^\times as the cochain (1+\alpha_{ij}t), and we have \begin{aligned} (1+\alpha_{ij}t) (1+\alpha_{jk}t) &= 1 + (\alpha_{ij}+\alpha_{jk})t + \alpha_{ij}\alpha_{jk}t^2 \\&= (1 + \alpha_{ik}t + \alpha_{ij}\alpha_{jk}t^2) \\&= (1+\alpha_{ik}t) \{1 + \alpha_{ij}\alpha_{jk}t^2). \end{aligned} Finally, let \delta_1 a=a\smile a where the cup product is taken in the sheaf of algebras \Omega.

Note that, if we denote by \bar{\smile} the cup product taken in the sheaf of algebras opposite to \Omega, i.e. defined on the level of cochains by (\alpha\bar{\smile}\beta)_{ijk}=\beta_{jk}\alpha_{ij}, we always have that a\bar{\smile}b=-b\smile a in cohomology.

Consequently, [a\smile a] = (a\smile a)-(a\bar{\smile}a) = 2a\smile a and \delta_1a=a\smile a=\frac12[a\smile a]. We thus recover, up to a factor of \frac12, the obstruction defined earlier in this talk.

## 3. Calculation of the secondary obstruction

Now suppose that a\smile a=0, so that we can find a cochain \beta=(\beta_{ij}) such that \delta\beta+\alpha\smile\alpha=0, i.e. \beta_{ik} = \beta_{ij}+\beta_{jk}+\alpha_{ij}\alpha_{jk}. Then (1+\alpha_{ij}t+\beta{ij}t^2) is a cocycle in \Omega_2^\times, and we can choose the cochain \beta to be a cocycle in {\mathscr{Q}}_2.

This cocycle can be lifted to \Omega_3^\times as the cochain (1+\alpha_{ij}t+\beta_{ij}t^2), and we have that \begin{aligned} &(1+\alpha_{ij}t+\beta_{ij}t^2) (1+\alpha_{jk}t+\beta_{jk}t^2) \\=\,\,& 1+ (\alpha_{ij}+\alpha_{jk})t + (\beta_{ij}+\beta_{jk}+\alpha_{ij}\alpha_{jk})t^2 + (\alpha_{ij}\beta_{jk}+\beta_{ij}\alpha_{jk})t^3 \\=\,\,& (1 + \alpha_{ik}t + \beta{ik}t^2) (1+ (\alpha_{ij}\beta_{jk}+\beta_{ij}\alpha_{jk})t^3). \end{aligned} The secondary obstruction of a is thus the cohomology class of the cocycle (\alpha_{ij}\beta_{jk}+\beta_{ij}\alpha_{jk})\in\operatorname{Z}^2(V_0;\Omega). This class depends on the choice of the cochain \beta: if we choose some other \beta'=\beta+\theta, where \Theta\in\operatorname{Z}^1(V_0;\Theta), then the cocycle is modified by \alpha\smile\theta+\theta\smile\alpha, and its class by an element of [a\smile\mathrm{H}^1(V_0;\Theta)]. We recover the Massey triple product (a,a,a) taken in the algebra \Omega, but with a slightly more restrictive indetermination.

We can try to calculate this secondary obstruction without leaving the sheaf \Theta, but the calculations are then much more complicated: we must take a cochain \beta=(\beta_{ij}) such that \delta\beta+\frac12[a\smile a]=0. Then the secondary obstruction of \alpha is the class of the cocycle [\alpha_{ij},\beta_{jk}] + \frac16[[\alpha_{ij},\alpha_{jk}],\alpha_ij+2\alpha_{jk}]. The calculation done in the sheaf of enveloping algebras \Omega can be generalised to obstructions of order r: we are led to determining, by induction, cochains \omega_r such that \begin{cases} \omega_1 = \alpha \\\delta\omega_r + \sum_{p+q=r}\omega_p\smile\omega_q = 0 \\1 + \sum_{1\leqslant p\leqslant r}\omega_p t^p \in \operatorname{C}^1(V_0;{\mathscr{Q}}_r) \end{cases}

## 4. Using spectral sequences

Let \varphi\colon V_0\to X be an arbitrary map, which gives rise to a spectral sequence of graded Lie algebras \mathrm{H}^\bullet(X;\mathbb{R}^\bullet\varphi\Theta) \Rightarrow \mathrm{H}^\bullet(V_0;\Theta). Let a \in \mathrm{H}^1(X;\varphi_*\Theta) \subset \mathrm{H}^1(V_0;\Theta). If the element -\frac12[a\smile a] \in \mathrm{H}^2(X;\varphi_*\Theta) = E_2^{2,0} is non-zero, but is the image under the differential \mathrm{d}_2 of the spectral sequence of an element b\in E_2^{0,1}, then the image of the secondary obstruction of a in E_\infty^{1,1} consists of the elements of the form [a,b]. In particular, if, for all b such that \mathrm{d}_2 b=-\frac12[a,a], we have that [a,b]\neq0, then the secondary obstruction is non-trivial.

However, if [a,b]=0 in E^{1,1}, then we can only say that the secondary obstruction comes from E_\infty^{2,0}, and if this group is non-zero, then we cannot conclude anything.

Proof. Let \alpha be a cocycle on V_0 representing the class a. The element b\in E_2^{0,1} can be represented by a cochain \beta = (\beta_{ij}) \in \operatorname{C}^1(V_0;\Theta) such that \delta\beta + \frac12[a\smile a] = 0. We thus obtain a cochain \beta' \in \operatorname{C}^1(V_0;\Omega) such that 1 + \alpha t + \beta' t^2 \in \operatorname{C}^1(V_0;{\mathscr{Q}}_2) by setting \beta'_{ij}=\beta_{ij}+\frac12\alpha_{ij}^2; this cochain satisfies \delta\beta'+\alpha\smile\alpha=0. But this new cochain represents, in the E_2^{0,1} term of the spectral sequence of the sheaf \Omega, the same element b as the cochain \beta, since it differs from it by a cochain that comes from X. The secondary obstruction is thus the class of the cocycle \alpha\smile\beta'+\beta'\smile\alpha, which represents in the E^{1,1} term of the spectral sequence the element [a,b].

This proposition allows us to construct non-trivial examples of secondary obstructions. Consider the group N of matrices of the form \begin{pmatrix} 1 & x & y \\0 & 1 & z \\0 & 0 & 1 \end{pmatrix} where x,y,z\in\mathbb{C}, and let Y=N/\Gamma, where \Gamma is the subgroup of N consisting of elements where x,y,z\in\mathbb{Z}+i\mathbb{Z}. Then Y is fibred over a complex torus of dimension two T^2\cong\mathbb{C}^2/\mathbb{Z}^4. We find non-trivial secondary obstruction elements in \mathrm{H}^1(V_0;\Theta), where V_0 is the product of Y with a projective line D. (We use the spectral sequence obtained by projecting onto T^2\times D). This variety has a “versal” deformation whose Zariski tangent space of the base B can be identified via the Spencer–Kodaira map \rho with \mathrm{H}^1(V_0;\Theta). Further, B has, at its base point b_0, a conic singularity of degree 3, whose equation is given by the secondary obstruction.

I do not know of any examples of non-trivial secondary obstructions on varieties V_0 that satisfy \mathrm{H}^0(V_0;\Theta)=0, but some very likely exist.

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1. [Trans.] The more common modern nomenclature is to simply call such an object a family of complex manifolds.↩︎

2. See the Appendix.↩︎